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Construction of Hilbert and Quot Schemes, and Its Application to the Construction of Picard Schemes (And Also a Sketch of Formal Schemes and Some Quotient Techniques)

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Construction of Hilbert and Quot Schemes, and Its Application to the Construction of Picard Schemes (And Also a Sketch of Formal Schemes and Some Quotient Techniques) Construction of Hilbert and Quot Schemes Nitin Nitsure School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai400005,India. e-mail: [email protected] Abstract This is an expository account of Grothendieck’s construction of Hilbert and Quot Schemes, following his talk ‘Techniques de construction et th´eor`emes d’existence en g´eom´etrie alg´ebriques IV : les sch´emas de Hilbert’, S´eminaire Bourbaki 221 (1960/61), together with further developments by Mumford and by Altman and Kleiman. Hilbert and Quot schemes are fundamental to modern Algebraic Geometry, in particular, for deformation theory and moduli constructions. These notes are based on a series of six lectures in the summer school ‘Advanced Basic Algebraic Geometry’, held at the Abdus Salam International Centre for Theoretical Physics, Trieste, in July 2003. Any scheme X defines a contravariant functor hX (called the functor of points of the scheme X) from the category of schemes to the category of sets, which associates to any scheme T the set Mor(T,X) of all morphisms from T to X. The scheme X can be recovered (up to a unique isomorphism) from hX by the Yoneda lemma. In fact, it is enough to know the restriction of this functor to the full subcategory consisting of affine schemes, in order to recover the scheme X. It is often easier to directly describe the functor hX than to give the scheme X. Such is typically the case with various parameter schemes and moduli schemes, or with various group-schemes over arbitrary bases, where we can directly define a contravariant functor F from the category of schemes to the category of sets which would be the functor of points of the scheme in question, without knowing in advance whether such a scheme indeed exists. This raises the problem of representability of contravariant functors from the cate- arXiv:math/0504590v1 [math.AG] 29 Apr 2005 gory of schemes to the category of sets. An important necessary condition for repre- sentability come from the fact that the functor hX satisfies descent under faithfully flat quasi-compact coverings. (Recall that descent for a set-valued functor F is the sheaf condition, which says that if (fi : Ui → U) is an open cover of U in the fpqc topology, then the diagram → of sets F (U) → i F (Ui) → i,j F (Ui ×U Uj) is exact.) The descent conditionQ is oftenQ easy to verify for a given functor F , but it is not a sufficient condition for representability. It is therefore a subtle and technically difficult problem in Algebraic Geometry to construct schemes which represent various important functors, such as moduli func- tors. Grothendieck addressed the issue by proving the representability of certain basic functors, namely, the Hilbert and Quot functors. The representing schemes 1 that he constructed, known as Hilbert schemes and Quot schemes, are the founda- tion for proving representability of most moduli functors (whether as schemes or as algebraic stacks). The techniques used by Grothendieck are based on the theories of descent and cohomology developed by him. In a sequence of talks in the Bourbaki seminar, collected under the title ‘Fondements de la G´eom´etrie Alg´ebriques’ (see [FGA]), he gave a sketch of the theory of descent, the construction of Hilbert and Quot schemes, and its application to the construction of Picard schemes (and also a sketch of formal schemes and some quotient techniques). The following notes give an expository account of the construction of Hilbert and Quot schemes. We assume that the reader is familiar with the basics of the language of schemes and cohomology, say at the level of chapters 2 and 3 of Hartshorne’s ‘Al- gebraic Geometry’ [H]. Some more advanced facts about flat morphisms (including the local criterion for flatness) that we need are available in Altman and Kleiman’s ‘Introduction to Grothendieck Duality Theory’ [A-K 1]. The lecture course by Vis- toli [V] on the theory of descent in this summer school contains in particular the background we need on descent. Certain advanced techniques of projective geom- etry, namely Castelnuovo-Mumford regularity and flattening stratification (to each of which we devote one lecture) are nicely given in Mumford’s ‘Lectures on Curves on an Algebraic Surface’ [M]. The book ‘Neron Models’ by Bosch, L¨utkebohmert, Raynaud [B-L-R] contains a quick exposition of descent, quot schemes, and Picard schemes. The reader of these lecture notes is strongly urged to read Grothendieck’s original presentation in [FGA]. 1 The Hilbert and Quot Functors The Functors HilbPn The main problem addressed in this series of lectures, in its simplest form, is as follows. If S is a locally noetherian scheme, a family of subschemes of Pn n n parametrised by S will mean a closed subscheme Y ⊂ PS = PZ × S such that Y is flat over S. If f : T → S is any morphism of locally noetherian schemes, then by ∗ −1 n pull-back we get a family f (Y ) = (id ×f) (Y ) ⊂ PT parametrised by T , from a family Y parametrised by S. This defines a contravariant functor HilbPn from the category of all locally noetherian schemes to the category of sets, which associates to any S the set of all such families n HilbPn (S)= {Y ⊂ PS | Y is flat over S} Question: Is the functor HilbPn representable? Grothendieck proved that this question has an affirmative answer, that is, there n exists a locally noetherian scheme HilbPn together with a family Z ⊂ PZ × HilbPn parametrised by HilbPn , such that any family Y over S is obtained as the pull-back of Z by a uniquely determined morphism ϕY : S → HilbPn . In other words, HilbPn is isomorphic to the functor Mor(−, HilbPn ). 2 r The Functors Quot⊕ OPn A family Y of subschemes of Pn parametrised by S is the same as a coherent quotient n n P sheaf q : OPS → OY on S, such that OY is flat over S. This way of looking at the functor HilbPn has the following fruitful generalisation. r Let r be any positive integer. A family of quotients of ⊕ OPn parametrised by a locally noetherian scheme S will mean a pair (F, q) consisting of n (i) a coherent sheaf F on PS which is flat over S, and n r n (ii) a surjective OPS -linear homomorphism of sheaves q : ⊕ OPS →F. Two such families (F, q)and (F, q) parametrised by S will be regarded as equivalent if there exists an isomorphism f : F→F ′ which takes q to q′, that is, the following diagram commutes. r q ⊕ OPn → F k ↓ f ′ r q ′ ⊕ OPn → F This is the same as the condition ker(q) = ker(q′). We will denote by hF, qi an equivalence class. If f : T → S is a morphism of locally noetherian schemes, then r n n n P P pulling back the quotient q : ⊕ OPS → F under id ×f : T → S defines a family ∗ r n ∗ f (q) : ⊕ OPT → f (F) over T , which makes sense as tensor product is right-exact and preserves flatness. The operation of pulling back respects equivalence of families, r therefore it gives rise to a contravariant functor Quot⊕ OPn from the category of all locally noetherian schemes to the category of sets, by putting r Quot⊕ OPn (S)= { All hF, qi parametrised by S} r It is immediate that the functor Quot⊕ OPn satisfies faithfully flat descent. It was proved by Grothendieck that in fact the above functor is representable on the cate- r gory of all locally noetherian schemes by a scheme Quot⊕ OPn . The Functors HilbX/S and QuotE/X/S n r The above functors HilbP and Quot⊕ OPn admit the following simple generalisa- tions. Let S be a noetherian scheme and let X → S be a finite type scheme over it. Let E be a coherent sheaf on X. Let SchS denote the category of all locally noetherian schemes over S. For any T → S in SchS, a family of quotients of E parametrised by T will mean a pair (F, q) consisting of (i) a coherent sheaf F on XT = X ×S T such that the schematic support of F is proper over T and F is flat over T , together with (ii) a surjective OXT -linear homomorphism of sheaves q : ET →F where ET is the pull-back of E under the projection XT → X. Two such families (F, q)and (F, q) parametrised by T will be regarded as equivalent if ker(q) = ker(q′)), and hF, qi will denote an equivalence class. Then as properness and flatness are preserved by base-change, and as tensor-product is right exact, the 3 pull-back of hF, qi under an S-morphism T ′ → T is well-defined, which gives a set-valued contravariant functor QuotE/X/S : SchS → Sets under which T 7→ { All hF, qi parametrised by T } When E = OX , the functor QuotOX /X/S : SchS → Sets associates to T the set of all closed subschemes Y ⊂ XT that are proper and flat over T . We denote this functor by HilbX/S. Note in particular that we have n n r r n HilbP = HilbP / Spec Z and Quot⊕ OPn = Quot⊕ OPn /P / Spec Z Z Z Z It is clear that the functors QuotE/X/S and HilbX/S satisfy faithfully flat descent, so it makes sense to pose the question of their representability. Stratification by Hilbert Polynomials Let X be a finite type scheme over a field k, together with a line bundle L. Recall that if F is a coherent sheaf on X whose support is proper over k, then the Hilbert polynomial Φ ∈ Q[λ] of F is defined by the function n i i ⊗m Φ(m)= χ(F (m)) = (−1) dimk H (X, F ⊗ L ) Xi=0 where the dimensions of the cohomologies are finite because of the coherence and properness conditions.
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